# On Application Oriented Fuzzy Numbers for Imprecise Investment Recommendations

## Abstract

**:**

## 1. Introduction

- The use of FNs in financial analysis only leads to averaging the imprecision risk,
- The application of OFNs in financial analysis may minimise imprecision risk.

## 2. Fuzzy Sets—Selected Facts

#### 2.1. Fuzzy Sets

#### 2.2. Fuzzy Numbers

**Theorem**

**1.**

**Definition**

**1.**

#### 2.3. Oriented Fuzzy Number

**Definition**

**2**

**.**For any monotonic sequence $\left(a,b,c,d\right)\subset \mathbb{R}$, the oriented fuzzy number OFN $\overleftrightarrow{\mathcal{L}}\left(a,b,c,d,{S}_{L},{E}_{L}\right)=\overleftrightarrow{\mathcal{L}}$ is a pair of an orientation $\overrightarrow{a,d}=\left(a,d\right)$ and a fuzzy set $\mathcal{L}\in \mathcal{F}(\mathbb{R})$ described by a membership function ${\mu}_{L}\left(\cdot |a,b,c,d,{S}_{L},{E}_{L}\right)\in {\left[0,1\right]}^{\mathbb{R}}$ given by the identity

**Definition**

**3**

**.**For any monotonic sequence $\left(a,b,c,d\right)\subset \mathbb{R}$, the trapezoidal OFN (TrOFN) $\overleftrightarrow{Tr}\left(a,b,c,d\right)=\overleftrightarrow{\mathcal{T}}$ is the pair of the orientation $\overrightarrow{a,d}=\left(a,d\right)$ and a fuzzy set $\mathcal{T}\in \mathcal{F}(\mathbb{R})$ determined by membership functions ${\mu}_{T}\in {\left[0,1\right]}^{\mathbb{R}}$ as follows

## 3. Oriented Present Value

- $\stackrel{\u02c7}{P}$ is a quoted price,
- $\left[{V}_{s},{V}_{e}\right]\subset {\mathbb{R}}^{+}$ is the set of all possible values of PV,
- $\left[{V}_{f},{V}_{l}\right]\subset \left[{V}_{s},{V}_{e}\right]$ is the set of all values that do not noticeably differ from the quoted price $\stackrel{\u02c7}{P}$.

**Example**

**1.**

## 4. Oriented Expected Discount Factor

**Example**

**2.**

- Expected RR${\overline{r}}_{s}$,
- CAPM directional factor${\beta}_{s}$,
- Variance${\sigma}_{s}^{2}$,
- Downside semi variance${\varsigma}_{s}^{2}$.

## 5. Investment Recommendations

- ${A}^{++}$ denotes the advice Buy suggesting that the expected price is well above the current quoted price,
- ${A}^{+}$ denotes the advice Accumulate suggesting that the expected price is above the current quoted price,
- ${A}^{0}$ denotes the advice Hold suggesting that the expected price is similar to the current quoted price,
- ${A}^{-}$ denotes the advice Reduce suggesting that the expected price is below the current quoted price,
- ${A}^{--}$ denotes the advice Sell suggesting that the expected price is well below the current quoted price.

## 6. The Profitability Criteria for Investments

**Example**

**3.**

#### 6.1. Sharpe Ratio

**Example**

**4.**

#### 6.2. Jensen’s Alpha

**Example**

**5.**

#### 6.3. Treynor Ratio

**Example**

**6.**

#### 6.4. Sortino Ratio

**Example**

**7.**

#### 6.5. Modiglianis’ Coefficient

#### 6.6. Roy’s Criterion

- $L$—a minimum acceptable RR,
- $\epsilon $—the probability of RR realisation below the minimum acceptable rate.

**Example**

**8.**

#### 6.7. Discussions

- Sharpe ratio and Sortino ratio used to maximise the premium per overall risk unit,
- Jensen’s alpha and Treynor ratio used to maximise the premium for market risk.
- Roy’s criterion used to minimise the probability of bearing the unacceptable loss.

## 7. Management of Investment Recommendation Set

- ${\tilde{\Lambda}}_{S,1}$—recommendations obtained with the use of the Sharpe ratio,
- ${\tilde{\Lambda}}_{S,2}$—recommendations obtained with the use of Jensen’s alpha,
- ${\tilde{\Lambda}}_{S,3}$—recommendations obtained with the use of the Treynor ratio,
- ${\tilde{\Lambda}}_{S,4}$—recommendations obtained with the use of the Sortino ratio,
- ${\tilde{\Lambda}}_{S,5}$—recommendations obtained with the use of Roy’s criterion.

**Example**

**10.**

- A weakly justified recommendation (WJR) ${\tilde{\Lambda}}_{S,WJR}$ defined as the union of such Pareto optimal recommendations, which are linked to the security $\widehat{S}$;
- A strongly justified recommendation (SJR) ${\tilde{\Lambda}}_{S,SJR}$ defined as the intersection of such Pareto optimal recommendations, which are linked to the security $\widehat{S}$.

**Example**

**11.**

- The investor bears almost all the responsibility for making an investment decision resulting from the Reduce and Hold recommendations,
- The advisor bears full responsibility for making an investment decision resulting from the Accumulate recommendations,
- The investor and the advisor share the responsibility among themselves for making the investment decision based on the Buy recommendation, however, the advisor bear approximately two-thirds of that responsibility.

## 8. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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Stock Company | $\mathbf{Present}\mathbf{Value}{\overleftrightarrow{\mathit{P}\mathit{V}}}_{\mathit{s}}$ | $\mathbf{Quoted}\mathbf{Price}{\stackrel{\u02c7}{\mathit{P}}}_{\mathit{s}}$ |
---|---|---|

ALR | $\overleftrightarrow{Tr}\left(27.42;27.30;27.00;26.84\right)$ | $27.00$ |

CCC | $\overleftrightarrow{Tr}\left(83.35;88.00;88.00;89.65\right)$ | $88.00$ |

CDR | $\overleftrightarrow{Tr}\left(271.50;271.50;276.30;276.30\right)$ | $277.00$ |

CPS | $\overleftrightarrow{Tr}\left(26.42;26.60;27.04;27.34\right)$ | $27.20$ |

DNP | $\overleftrightarrow{Tr}\left(155.00;155.00;155.10;157.30\right)$ | $155.30$ |

JSW | $\overleftrightarrow{Tr}\left(18.60;19.36;20.14;20.14\right)$ | $20.32$ |

KGH | $\overleftrightarrow{Tr}\left(91.78;93.60;93.70;94.90\right)$ | $94.24$ |

LTS | $\overleftrightarrow{Tr}\left(83.88;83.40;81.16;80.26\right)$ | $81.44$ |

LPP | $\overleftrightarrow{Tr}\left(8205.00;8380.00;8395.00;8460.00\right)$ | $8385.00$ |

MBK | $\overleftrightarrow{Tr}\left(367.00;366.00;359.80;357.00\right)$ | $359.00$ |

OPL | $\overleftrightarrow{Tr}\left(7.01;7.05;7.20;7.35\right)$ | $7.17$ |

PEO | $\overleftrightarrow{Tr}\left(97.22;97.70;98.20;98.66\right)$ | $98.20$ |

PGE | $\overleftrightarrow{Tr}\left(7.08;7.15;7.30;7.40\right)$ | $7.30$ |

PGN | $\overleftrightarrow{Tr}\left(3.91;3.88;3.86;3.82\right)$ | $3.87$ |

PKN | $\overleftrightarrow{Tr}\left(83.22;83.00;81.62;81.18\right)$ | $81.90$ |

PKO | $\overleftrightarrow{Tr}\left(34.59;34.68;34.90;35.26\right)$ | $34.93$ |

PLY | $\overleftrightarrow{Tr}\left(35.82;35.94;36.76;37.20\right)$ | $36.70$ |

PZU | $\overleftrightarrow{Tr}\left(40.72;40.73;40.89;41.11\right)$ | $40.88$ |

SPL | $\overleftrightarrow{Tr}\left(276.20;278.00;281.80;283.80\right)$ | $287.00$ |

TPE | $\overleftrightarrow{Tr}\left(1.51;1.53;1.56;1.56\right)$ | $1.56$ |

Stock Company | $\mathbf{Expected}\mathbf{Return}\mathbf{Rate}{\overline{\mathit{r}}}_{\mathit{s}}$ | $\mathbf{CAPM}\mathbf{Factor}{\mathit{\beta}}_{\mathit{s}}$ | $\mathbf{Variance}{\mathit{\sigma}}_{\mathit{s}}^{2}$ | $\mathbf{Downside}\mathbf{Semi}\mathbf{Variance}{\mathit{\varsigma}}_{\mathit{s}}^{2}$ | $\mathbf{EDF}{\overline{\mathit{v}}}_{\mathit{s}}$ | $\mathbf{OEDF}{\overleftrightarrow{\mathit{V}}}_{\mathit{s}}$ |
---|---|---|---|---|---|---|

ALR | $0.0263$ | $1.706$ | $0.000031$ | $0.000018$ | $0.9744$ | $\overleftrightarrow{Tr}\left(0.9896;0.9852;0.9744;0.9686\right)$ |

CCC | $0.0367$ | $2.160$ | $0.000048$ | $0.000026$ | $0.9646$ | $\overleftrightarrow{Tr}\left(0.9136;0.9646;0.9646;0.9827\right)$ |

CDR | $0.2490$ | $9.925$ | $0.000311$ | $0.000189$ | $0.8006$ | $\overleftrightarrow{Tr}\left(0.7847;0.7847;0.7986;0.7986\right)$ |

CPS | $0.0594$ | $3.852$ | $0.000093$ | $0.000050$ | $0.9439$ | $\overleftrightarrow{Tr}\left(0.9168;0.9231;0.9384;0.9488\right)$ |

DNP | $0.0672$ | $3.465$ | $0.000011$ | $0.000006$ | $0.9370$ | $\overleftrightarrow{Tr}\left(0.9352;0.9352;0.9358;0.9491\right)$ |

JSW | $0.0199$ | $-0.598$ | $0.000016$ | $0.000010$ | $0.9805$ | $\overleftrightarrow{Tr}\left(0.8975;0.9342;0.9718;0.9718\right)$ |

KGH | $0.0567$ | $2.699$ | $0.000063$ | $0.000039$ | $0.9463$ | $\overleftrightarrow{Tr}\left(0.9216;0.9399;0.9409;0.9529\right)$ |

LTS | $0.1054$ | $3.643$ | $0.000161$ | $0.000092$ | $0.9047$ | $\overleftrightarrow{Tr}\left(0.9318;0.9265;0.9016;0.8916\right)$ |

LPP | $0.0872$ | $1.958$ | $0.000126$ | $0.000071$ | $0.9198$ | $\overleftrightarrow{Tr}\left(0.9001;0.9193;0.9209;0.9280\right)$ |

MBK | $0.0674$ | $6.243$ | $0.000097$ | $0.000059$ | $0.9369$ | $\overleftrightarrow{Tr}\left(0.9578;0.9552;0.9390;0.9317\right)$ |

OPL | $0.0278$ | $1.9406$ | $0.000028$ | $0.000017$ | $0.9730$ | $\overleftrightarrow{Tr}\left(0.9513;0.9567;0.9771;0.9974\right)$ |

PEO | $0.0459$ | $1.348$ | $0.000068$ | $0.000036$ | $0.9561$ | $\overleftrightarrow{Tr}\left(0.9466;0.9512;0.9561;0.9606\right)$ |

PGE | $0.0674$ | $4.392$ | $0.000099$ | $0.000071$ | $0.9369$ | $\overleftrightarrow{Tr}\left(0.9087;0.9177;0.9369;0.9497\right)$ |

PGN | $0.0751$ | $3.976$ | $0.000109$ | $0.000065$ | $0.9302$ | $\overleftrightarrow{Tr}\left(0.9398;0.9326;0.9278;0.9182\right)$ |

PKN | $0.0408$ | $2.674$ | $0.000049$ | $0.000029$ | $0.9608$ | $\overleftrightarrow{Tr}\left(0.9763;0.9737;0.9575;0.9524\right)$ |

PKO | $0.1974$ | $5.463$ | $0.000216$ | $0.000099$ | $0.8351$ | $\overleftrightarrow{Tr}\left(0.8270;0.8291;0.8344;0.8430\right)$ |

PLY | $0.2607$ | $6.2156$ | $0.000589$ | $0.000312$ | $0.7932$ | $\overleftrightarrow{Tr}\left(0.7742;0.7768;0.7945;0.8040\right)$ |

PZU | $0.1952$ | $5.541$ | $0.000301$ | $0.000181$ | $0.8367$ | $\overleftrightarrow{Tr}\left(0.8334;0.8336;0.8369;0.8414\right)$ |

SPL | $0.3001$ | $8.867$ | $0.000563$ | $0.000391$ | $0.7692$ | $\overleftrightarrow{Tr}\left(0.7403;0.7451;0.7553;0.7606\right)$ |

TPE | $0.0432$ | $2.991$ | $0.000056$ | $0.000035$ | $0.9586$ | $\overleftrightarrow{Tr}\left(0.9279;0.9402;0.9586;0.9586\right)$ |

Stock Company | $\mathbf{OEDF}{\overleftrightarrow{\mathit{V}}}_{\mathit{s}}$ | $\mathbf{SPT}{\mathit{H}}_{\mathit{s}}$ |
---|---|---|

ALR | $\overleftrightarrow{Tr}\left(0.9896;0.9852;0.9744;0.9686\right)$ | $0.9790$ |

CCC | $\overleftrightarrow{Tr}\left(0.9136;0.9646;0.9646;0.9827\right)$ | $0.9758$ |

CDR | $\overleftrightarrow{Tr}\left(0.7847;0.7847;0.7986;0.7986\right)$ | $0.9509$ |

CPS | $\overleftrightarrow{Tr}\left(0.9168;0.9231;0.9384;0.9488\right)$ | $0.9694$ |

DNP | $\overleftrightarrow{Tr}\left(0.9352;0.9352;0.9358;0.9491\right)$ | $0.9845$ |

JSW | $\overleftrightarrow{Tr}\left(0.8975;0.9342;0.9718;0.9718\right)$ | $0.9828$ |

KGH | $\overleftrightarrow{Tr}\left(0.9216;0.9399;0.9409;0.9529\right)$ | $0.9734$ |

LTS | $\overleftrightarrow{Tr}\left(0.9318;0.9265;0.9016;0.8916\right)$ | $0.9623$ |

LPP | $\overleftrightarrow{Tr}\left(0.9001;0.9193;0.9209;0.9280\right)$ | $0.9657$ |

MBK | $\overleftrightarrow{Tr}\left(0.9578;0.9552;0.9390;0.9317\right)$ | $0.9689$ |

OPL | $\overleftrightarrow{Tr}\left(0.9513;0.9567;0.9771;0.9974\right)$ | $0.9797$ |

PEO | $\overleftrightarrow{Tr}\left(0.9466;0.9512;0.9561;0.9606\right)$ | $0.9727$ |

PGE | $\overleftrightarrow{Tr}\left(0.9087;0.9177;0.9369;0.9497\right)$ | $0.9686$ |

PGN | $\overleftrightarrow{Tr}\left(0.9398;0.9326;0.9278;0.9182\right)$ | $0.9675$ |

PKN | $\overleftrightarrow{Tr}\left(0.9763;0.9737;0.9575;0.9524\right)$ | $0.9756$ |

PKO | $\overleftrightarrow{Tr}\left(0.8270;0.8291;0.8344;0.8430\right)$ | $0.9576$ |

PLY | $\overleftrightarrow{Tr}\left(0.7742;0.7768;0.7945;0.8040\right)$ | $0.9362$ |

PZU | $\overleftrightarrow{Tr}\left(0.8334;0.8336;0.8369;0.8414\right)$ | $0.9516$ |

SPL | $\overleftrightarrow{Tr}\left(0.7403;0.7451;0.7553;0.7606\right)$ | $0.9461$ |

TPE | $\overleftrightarrow{Tr}\left(0.9279;0.9402;0.9586;0.9586\right)$ | $0.9745$ |

Recommendation Choice Function | |||||||
---|---|---|---|---|---|---|---|

Stock Company | ${\mathit{A}}^{--}$ | ${\mathit{A}}^{-}$ | ${\mathit{A}}^{0}$ | ${\mathit{A}}^{+}$ | ${\mathit{A}}^{++}$ | Energy Measure | Entropy Measure |

ALR | $0$ | $1$ | $1$ | $1$ | $0$ | $3$ | $0$ |

CCC | $0$ | $0.3812$ | $0.3812$ | $1$ | $0.6188$ | $2.3812$ | $0.2966$ |

CDR | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

CPS | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

DNP | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

JSW | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

KGH | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

LTS | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

LPP | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

MBK | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

OPL | $0$ | $0.8719$ | $0.8719$ | $1$ | $0.1281$ | $2.8719$ | $0.0833$ |

PEO | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PGE | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PGN | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PKN | $0$ | $0.2692$ | $0.2692$ | $1$ | $0.7308$ | $2.2692$ | $0.1926$ |

PKO | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PLY | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PZU | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

SPL | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

TPE | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

Stock Company | $\mathbf{OEDF}{\overleftrightarrow{\mathit{V}}}_{\mathit{s}}$ | $\mathbf{SPT}{\mathit{H}}_{\mathit{s}}$ |
---|---|---|

ALR | $\overleftrightarrow{Tr}\left(0.9896;0.9852;0.9744;0.9686\right)$ | $0.9720$ |

CCC | $\overleftrightarrow{Tr}\left(0.9136;0.9646;0.9646;0.9827\right)$ | $0.9667$ |

CDR | $\overleftrightarrow{Tr}\left(0.7847;0.7847;0.7986;0.7986\right)$ | $0.8837$ |

CPS | $\overleftrightarrow{Tr}\left(0.9168;0.9231;0.9384;0.9488\right)$ | $0.9473$ |

DNP | $\overleftrightarrow{Tr}\left(0.9352;0.9352;0.9358;0.9491\right)$ | $0.9517$ |

JSW | $\overleftrightarrow{Tr}\left(0.8975;0.9342;0.9718;0.9718\right)$ | $1.0000$ |

KGH | $\overleftrightarrow{Tr}\left(0.9216;0.9399;0.9409;0.9529\right)$ | $0.9604$ |

LTS | $\overleftrightarrow{Tr}\left(0.9318;0.9265;0.9016;0.8916\right)$ | $0.9496$ |

LPP | $\overleftrightarrow{Tr}\left(0.9001;0.9193;0.9209;0.9280\right)$ | $0.9690$ |

MBK | $\overleftrightarrow{Tr}\left(0.9578;0.9552;0.9390;0.9317\right)$ | $0.9212$ |

OPL | $\overleftrightarrow{Tr}\left(0.9513;0.9567;0.9771;0.9974\right)$ | $0.9692$ |

PEO | $\overleftrightarrow{Tr}\left(0.9466;0.9512;0.9561;0.9606\right)$ | $0.9762$ |

PGE | $\overleftrightarrow{Tr}\left(0.9087;0.9177;0.9369;0.9497\right)$ | $0.9413$ |

PGN | $\overleftrightarrow{Tr}\left(0.9398;0.9326;0.9278;0.9182\right)$ | $0.9459$ |

PKN | $\overleftrightarrow{Tr}\left(0.9763;0.9737;0.9575;0.9524\right)$ | $0.9607$ |

PKO | $\overleftrightarrow{Tr}\left(0.8270;0.8291;0.8344;0.8430\right)$ | $0.9296$ |

PLY | $\overleftrightarrow{Tr}\left(0.7742;0.7768;0.7945;0.8040\right)$ | $0.9215$ |

PZU | $\overleftrightarrow{Tr}\left(0.8334;0.8336;0.8369;0.8414\right)$ | $0.9287$ |

SPL | $\overleftrightarrow{Tr}\left(0.7403;0.7451;0.7553;0.7606\right)$ | $0.8942$ |

TPE | $\overleftrightarrow{Tr}\left(0.9279;0.9402;0.9586;0.9586\right)$ | $0.9570$ |

Recommendation Choice Function | |||||||
---|---|---|---|---|---|---|---|

Stock Company | ${\mathit{A}}^{--}$ | ${\mathit{A}}^{-}$ | ${\mathit{A}}^{0}$ | ${\mathit{A}}^{+}$ | ${\mathit{A}}^{++}$ | Energy Measure | Entropy Measure |

ALR | $0.4138$ | $1$ | $0.5862$ | $0.5862$ | $0$ | $2.5862$ | $0.3303$ |

CCC | $0$ | $0.8840$ | $0.8840$ | $1$ | $0.1160$ | $2.8840$ | $0.0748$ |

CDR | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

CPS | $0$ | $0.1442$ | $0.1442$ | $1$ | $0.8558$ | $2.1442$ | $0.0947$ |

DNP | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

JSW | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

KGH | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

LTS | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

LPP | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

MBK | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

OPL | $0$ | $1$ | $1$ | $1$ | $0$ | $3$ | $0$ |

PEO | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PGE | $0$ | $0.6563$ | $0.6563$ | $1$ | $0.3437$ | $2.6563$ | $0.2598$ |

PGN | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PKN | $0$ | $1$ | $1$ | $1$ | $0$ | $3$ | $0$ |

PKO | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PLY | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PZU | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

SPL | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

TPE | $0$ | $1$ | $1$ | $1$ | $0$ | $3$ | $0$ |

Stock Company | $\mathbf{OEDF}{\overleftrightarrow{\mathit{V}}}_{\mathit{s}}$ | $\mathbf{SPT}{\mathit{H}}_{\mathit{s}}$ |
---|---|---|

ALR | $\overleftrightarrow{Tr}\left(0.9896;0.9852;0.9744;0.9686\right)$ | $0.9545$ |

CCC | $\overleftrightarrow{Tr}\left(0.9136;0.9646;0.9646;0.9827\right)$ | $0.9401$ |

CDR | $\overleftrightarrow{Tr}\left(0.7847;0.7847;0.7986;0.7986\right)$ | $0.7283$ |

CPS | $\overleftrightarrow{Tr}\left(0.9168;0.9231;0.9384;0.9488\right)$ | $0.9029$ |

DNP | $\overleftrightarrow{Tr}\left(0.9352;0.9352;0.9358;0.9491\right)$ | $0.9005$ |

JSW | $\overleftrightarrow{Tr}\left(0.8975;0.9342;0.9718;0.9718\right)$ | $0.9877$ |

KGH | $\overleftrightarrow{Tr}\left(0.9216;0.9399;0.9409;0.9529\right)$ | $0.9171$ |

LTS | $\overleftrightarrow{Tr}\left(0.9318;0.9265;0.9016;0.8916\right)$ | $0.8689$ |

LPP | $\overleftrightarrow{Tr}\left(0.9001;0.9193;0.9209;0.9280\right)$ | $0.8995$ |

MBK | $\overleftrightarrow{Tr}\left(0.9578;0.9552;0.9390;0.9317\right)$ | $0.8730$ |

OPL | $\overleftrightarrow{Tr}\left(0.9513;0.9567;0.9771;0.9974\right)$ | $0.9505$ |

PEO | $\overleftrightarrow{Tr}\left(0.9466;0.9512;0.9561;0.9606\right)$ | $0.9410$ |

PGE | $\overleftrightarrow{Tr}\left(0.9087;0.9177;0.9369;0.9497\right)$ | $0.8910$ |

PGN | $\overleftrightarrow{Tr}\left(0.9398;0.9326;0.9278;0.9182\right)$ | $0.8891$ |

PKN | $\overleftrightarrow{Tr}\left(0.9763;0.9737;0.9575;0.9524\right)$ | $0.9309$ |

PKO | $\overleftrightarrow{Tr}\left(0.8270;0.8291;0.8344;0.8430\right)$ | $0.7901$ |

PLY | $\overleftrightarrow{Tr}\left(0.7742;0.7768;0.7945;0.8040\right)$ | $0.7472$ |

PZU | $\overleftrightarrow{Tr}\left(0.8334;0.8336;0.8369;0.8414\right)$ | $0.7909$ |

SPL | $\overleftrightarrow{Tr}\left(0.7403;0.7451;0.7553;0.7606\right)$ | $0.7088$ |

TPE | $\overleftrightarrow{Tr}\left(0.9279;0.9402;0.9586;0.9586\right)$ | $0.9254$ |

Recommendation Choice Function | |||||||
---|---|---|---|---|---|---|---|

Stock Company | ${\mathit{A}}^{--}$ | ${\mathit{A}}^{-}$ | ${\mathit{A}}^{0}$ | ${\mathit{A}}^{+}$ | ${\mathit{A}}^{++}$ | Energy Measure | Entropy Measure |

ALR | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

CCC | $0.4804$ | 1 | $0.5196$ | $0.5196$ | $0$ | $2.5196$ | $0.4050$ |

CDR | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

CPS | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

DNP | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

JSW | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

KGH | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

LTS | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

LPP | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

MBK | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

OPL | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

PEO | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

PGE | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

PGN | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

PKN | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

PKO | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

PLY | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

PZU | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

SPL | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

TPE | $1$ | $1$ | $0$ | $0$ | $0$ | $2$ | $0$ |

Stock Company | $\mathbf{OEDF}{\overleftrightarrow{\mathit{V}}}_{\mathit{s}}$ | $\mathbf{SPT}{\mathit{H}}_{\mathit{s}}$ |
---|---|---|

ALR | $\overleftrightarrow{Tr}\left(0.9896;0.9852;0.9744;0.9686\right)$ | $0.9793$ |

CCC | $\overleftrightarrow{Tr}\left(0.9136;0.9646;0.9646;0.9827\right)$ | $0.9766$ |

CDR | $\overleftrightarrow{Tr}\left(0.7847;0.7847;0.7986;0.7986\right)$ | $0.9507$ |

CPS | $\overleftrightarrow{Tr}\left(0.9168;0.9231;0.9384;0.9488\right)$ | $0.9706$ |

DNP | $\overleftrightarrow{Tr}\left(0.9352;0.9352;0.9358;0.9491\right)$ | $0.9848$ |

JSW | $\overleftrightarrow{Tr}\left(0.8975;0.9342;0.9718;0.9718\right)$ | $0.9826$ |

KGH | $\overleftrightarrow{Tr}\left(0.9216;0.9399;0.9409;0.9529\right)$ | $0.9731$ |

LTS | $\overleftrightarrow{Tr}\left(0.9318;0.9265;0.9016;0.8916\right)$ | $0.9630$ |

LPP | $\overleftrightarrow{Tr}\left(0.9001;0.9193;0.9209;0.9280\right)$ | $0.9665$ |

MBK | $\overleftrightarrow{Tr}\left(0.9578;0.9552;0.9390;0.9317\right)$ | $0.9687$ |

OPL | $\overleftrightarrow{Tr}\left(0.9513;0.9567;0.9771;0.9974\right)$ | $0.9796$ |

PEO | $\overleftrightarrow{Tr}\left(0.9466;0.9512;0.9561;0.9606\right)$ | $0.9738$ |

PGE | $\overleftrightarrow{Tr}\left(0.9087;0.9177;0.9369;0.9497\right)$ | $0.9665$ |

PGN | $\overleftrightarrow{Tr}\left(0.9398;0.9326;0.9278;0.9182\right)$ | $0.9676$ |

PKN | $\overleftrightarrow{Tr}\left(0.9763;0.9737;0.9575;0.9524\right)$ | $0.9757$ |

PKO | $\overleftrightarrow{Tr}\left(0.8270;0.8291;0.8344;0.8430\right)$ | $0.9619$ |

PLY | $\overleftrightarrow{Tr}\left(0.7742;0.7768;0.7945;0.8040\right)$ | $0.9394$ |

PZU | $\overleftrightarrow{Tr}\left(0.8334;0.8336;0.8369;0.8414\right)$ | $0.9516$ |

SPL | $\overleftrightarrow{Tr}\left(0.7403;0.7451;0.7553;0.7606\right)$ | $0.9334$ |

TPE | $\overleftrightarrow{Tr}\left(0.9279;0.9402;0.9586;0.9586\right)$ | $0.9741$ |

Recommendation Choice Function | |||||||
---|---|---|---|---|---|---|---|

Stock Company | ${\mathit{A}}^{--}$ | ${\mathit{A}}^{-}$ | ${\mathit{A}}^{0}$ | ${\mathit{A}}^{+}$ | ${\mathit{A}}^{++}$ | Energy Measure | Entropy Measure |

ALR | $0$ | $1$ | $1$ | $1$ | $0$ | $3$ | $0$ |

CCC | $0$ | $0.3370$ | $0.3370$ | $1$ | $0.663$ | $2.3370$ | $0.0228$ |

CDR | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

CPS | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

DNP | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

JSW | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

KGH | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

LTS | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

LPP | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

MBK | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

OPL | $0$ | $0.8769$ | $0.8769$ | $1$ | $0.1231$ | $2.8769$ | $0.0798$ |

PEO | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PGE | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PGN | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PKN | $0$ | $0.2308$ | $0.2308$ | $1$ | $0.7692$ | $2.2308$ | $0.1607$ |

PKO | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PLY | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PZU | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

SPL | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

TPE | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

Stock Company | $\mathbf{OEDF}{\overleftrightarrow{\mathit{V}}}_{\mathit{s}}$ | $\mathbf{SPT}{\mathit{H}}_{\mathit{s}}$ |
---|---|---|

ALR | $\overleftrightarrow{Tr}\left(0.9896;0.9852;0.9744;0.9686\right)$ | $0.9836$ |

CCC | $\overleftrightarrow{Tr}\left(0.9136;0.9646;0.9646;0.9827\right)$ | $0.9815$ |

CDR | $\overleftrightarrow{Tr}\left(0.7847;0.7847;0.7986;0.7986\right)$ | $0.9649$ |

CPS | $\overleftrightarrow{Tr}\left(0.9168;0.9231;0.9384;0.9488\right)$ | $0.9772$ |

DNP | $\overleftrightarrow{Tr}\left(0.9352;0.9352;0.9358;0.9491\right)$ | $0.9873$ |

JSW | $\overleftrightarrow{Tr}\left(0.8975;0.9342;0.9718;0.9718\right)$ | $0.9861$ |

KGH | $\overleftrightarrow{Tr}\left(0.9216;0.9399;0.9409;0.9529\right)$ | $0.9799$ |

LTS | $\overleftrightarrow{Tr}\left(0.9318;0.9265;0.9016;0.8916\right)$ | $0.9725$ |

LPP | $\overleftrightarrow{Tr}\left(0.9001;0.9193;0.9209;0.9280\right)$ | $0.9748$ |

MBK | $\overleftrightarrow{Tr}\left(0.9578;0.9552;0.9390;0.9317\right)$ | $0.9769$ |

OPL | $\overleftrightarrow{Tr}\left(0.9513;0.9567;0.9771;0.9974\right)$ | $0.9841$ |

PEO | $\overleftrightarrow{Tr}\left(0.9466;0.9512;0.9561;0.9606\right)$ | $0.9794$ |

PGE | $\overleftrightarrow{Tr}\left(0.9087;0.9177;0.9369;0.9497\right)$ | $0.9767$ |

PGN | $\overleftrightarrow{Tr}\left(0.9398;0.9326;0.9278;0.9182\right)$ | $0.9760$ |

PKN | $\overleftrightarrow{Tr}\left(0.9763;0.9737;0.9575;0.9524\right)$ | $0.9814$ |

PKO | $\overleftrightarrow{Tr}\left(0.8270;0.8291;0.8344;0.8430\right)$ | $0.9694$ |

PLY | $\overleftrightarrow{Tr}\left(0.7742;0.7768;0.7945;0.8040\right)$ | $0.9548$ |

PZU | $\overleftrightarrow{Tr}\left(0.8334;0.8336;0.8369;0.8414\right)$ | $0.9653$ |

SPL | $\overleftrightarrow{Tr}\left(0.7403;0.7451;0.7553;0.7606\right)$ | $0.9557$ |

TPE | $\overleftrightarrow{Tr}\left(0.9279;0.9402;0.9586;0.9586\right)$ | $0.9806$ |

Recommendation Choice Function | |||||||
---|---|---|---|---|---|---|---|

Stock Company | ${\mathit{A}}^{--}$ | ${\mathit{A}}^{-}$ | ${\mathit{A}}^{0}$ | ${\mathit{A}}^{+}$ | ${\mathit{A}}^{++}$ | Energy Measure | Entropy Measure |

ALR | $0$ | $1$ | $1$ | $1$ | $0$ | $3$ | $0$ |

CCC | $0$ | $0.0663$ | $0.0663$ | $1$ | $0.9337$ | $2.0663$ | $0.0414$ |

CDR | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

CPS | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

DNP | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

JSW | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

KGH | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

LTS | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

LPP | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

MBK | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

OPL | $0$ | $1$ | $1$ | $1$ | $0$ | $3$ | $0$ |

PEO | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PGE | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PGN | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PKN | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PKO | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PLY | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

PZU | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

SPL | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

TPE | $0$ | $0$ | $0$ | $1$ | $1$ | $2$ | $0$ |

Stock Company | Pareto Optimum |
---|---|

ALR | $\left\{{\tilde{\Lambda}}_{ALR,3}\right\}$ |

CCC | $\left\{{\tilde{\Lambda}}_{CCC,4},{\tilde{\Lambda}}_{CCC,5}\right\}$ |

CDR | $\left\{{\tilde{\Lambda}}_{CDR,1},{\tilde{\Lambda}}_{CDR,2},{\tilde{\Lambda}}_{CDR,3},{\tilde{\Lambda}}_{CDR,4},{\tilde{\Lambda}}_{CDR,5}\right\}$ |

CPS | $\left\{{\tilde{\Lambda}}_{CPS,1},{\tilde{\Lambda}}_{CPS,3},{\tilde{\Lambda}}_{CPS,4},{\tilde{\Lambda}}_{CPS,5}\right\}$ |

DNP | $\left\{{\tilde{\Lambda}}_{DNP,1},{\tilde{\Lambda}}_{DNP,2},{\tilde{\Lambda}}_{DNP,3},{\tilde{\Lambda}}_{DNP,4},{\tilde{\Lambda}}_{DNP,5}\right\}$ |

JSW | $\left\{{\tilde{\Lambda}}_{JSW,1},{\tilde{\Lambda}}_{JSW,2},{\tilde{\Lambda}}_{JSW,3},{\tilde{\Lambda}}_{JSW,4},{\tilde{\Lambda}}_{JSW,5}\right\}$ |

KGH | $\left\{{\tilde{\Lambda}}_{KGH,1},{\tilde{\Lambda}}_{KGH,2},{\tilde{\Lambda}}_{KGH,3},{\tilde{\Lambda}}_{KGH,4},{\tilde{\Lambda}}_{KGH,5}\right\}$ |

LTS | $\left\{{\tilde{\Lambda}}_{LTS,1},{\tilde{\Lambda}}_{LTS,2},{\tilde{\Lambda}}_{LTS,3},{\tilde{\Lambda}}_{LTS,4},{\tilde{\Lambda}}_{LTS,5}\right\}$ |

LPP | $\left\{{\tilde{\Lambda}}_{LPP,1},{\tilde{\Lambda}}_{LPP,2},{\tilde{\Lambda}}_{LPP,3},{\tilde{\Lambda}}_{LPP,4},{\tilde{\Lambda}}_{LPP,5}\right\}$ |

MBK | $\left\{{\tilde{\Lambda}}_{MBK,1},{\tilde{\Lambda}}_{MBK,2},{\tilde{\Lambda}}_{MBK,3},{\tilde{\Lambda}}_{MBK,4},{\tilde{\Lambda}}_{MBK,5}\right\}$ |

OPL | $\left\{{\tilde{\Lambda}}_{OPL,3}\right\}$ |

PEO | $\left\{{\tilde{\Lambda}}_{PEO,1},{\tilde{\Lambda}}_{PEO,2},{\tilde{\Lambda}}_{PEO,3},{\tilde{\Lambda}}_{PEO,4},{\tilde{\Lambda}}_{PEO,5}\right\}$ |

PGE | $\left\{{\tilde{\Lambda}}_{PGE,1},{\tilde{\Lambda}}_{PGE,3},{\tilde{\Lambda}}_{PGE,4},{\tilde{\Lambda}}_{PGE,5}\right\}$ |

PGN | $\left\{{\tilde{\Lambda}}_{PGN,1},{\tilde{\Lambda}}_{PGN,2},{\tilde{\Lambda}}_{PGN,3},{\tilde{\Lambda}}_{PGN,4},{\tilde{\Lambda}}_{PGN,5}\right\}$ |

PKN | $\left\{{\tilde{\Lambda}}_{PKN,3},{\tilde{\Lambda}}_{PKN,5}\right\}$ |

PKO | $\left\{{\tilde{\Lambda}}_{PKO,1},{\tilde{\Lambda}}_{PKO,2},{\tilde{\Lambda}}_{PKO,3},{\tilde{\Lambda}}_{PKO,4},{\tilde{\Lambda}}_{PKO,5}\right\}$ |

PLY | $\left\{{\tilde{\Lambda}}_{PLY,1},{\tilde{\Lambda}}_{PLY,2},{\tilde{\Lambda}}_{PLY,3},{\tilde{\Lambda}}_{PLY,4},{\tilde{\Lambda}}_{PLY,5}\right\}$ |

PZU | $\left\{{\tilde{\Lambda}}_{PZU,1},{\tilde{\Lambda}}_{PZU,2},{\tilde{\Lambda}}_{PZU,3},{\tilde{\Lambda}}_{PZU,4},{\tilde{\Lambda}}_{PZU,5}\right\}$ |

SPL | $\left\{{\tilde{\Lambda}}_{SPL,1},{\tilde{\Lambda}}_{SPL,2},{\tilde{\Lambda}}_{SPL,3},{\tilde{\Lambda}}_{SPL,4},{\tilde{\Lambda}}_{SPL,5}\right\}$ |

TPE | $\left\{{\tilde{\Lambda}}_{TPE,1},{\tilde{\Lambda}}_{TPE,2},{\tilde{\Lambda}}_{TPE,3},{\tilde{\Lambda}}_{TPE,4},{\tilde{\Lambda}}_{TPE,5}\right\}$ |

Criterion | ${\mathit{A}}^{--}$ | ${\mathit{A}}^{-}$ | ${\mathit{A}}^{0}$ | ${\mathit{A}}^{+}$ | ${\mathit{A}}^{++}$ | $\mathit{d}\left({\tilde{\mathit{\Lambda}}}_{\mathit{A}\mathit{L}\mathit{R},\mathit{i}}\right)$ | $\mathit{e}\left({\tilde{\mathit{\Lambda}}}_{\mathit{A}\mathit{L}\mathit{R},\mathit{i}}\right)$ |
---|---|---|---|---|---|---|---|

${\tilde{\Lambda}}_{ALR,1}$ | $0$ | $1$ | $1$ | $1$ | $0$ | $3$ | $0$ |

${\tilde{\Lambda}}_{ALR,2}$ | $0.4138$ | $1$ | $0.5862$ | $0.5862$ | $0$ | $2.5862$ | $0.3303$ |

${\tilde{\Lambda}}_{ALR,3}$ | $1$ | 1 | $0$ | $0$ | $0$ | $2$ | $0$ |

${\tilde{\Lambda}}_{ALR,4}$ | $0$ | $1$ | $1$ | $1$ | $0$ | $3$ | $0$ |

${\tilde{\Lambda}}_{ALR,5}$ | $0$ | $1$ | $1$ | $1$ | $0$ | $3$ | $0$ |

${\tilde{\Lambda}}_{ALR,WJR}$ | $1$ | 1 | $0$ | $0$ | $0$ | ||

${\tilde{\Lambda}}_{ALR,SJR}$ | $1$ | 1 | $0$ | $0$ | $0$ |

Criterion | ${\mathit{A}}^{--}$ | ${\mathit{A}}^{-}$ | ${\mathit{A}}^{0}$ | ${\mathit{A}}^{+}$ | ${\mathit{A}}^{++}$ | $\mathit{d}\left({\tilde{\mathit{\Lambda}}}_{\mathit{C}\mathit{C}\mathit{C},\mathit{i}}\right)$ | $\mathit{e}\left({\tilde{\mathit{\Lambda}}}_{\mathit{C}\mathit{C}\mathit{C},\mathit{i}}\right)$ |
---|---|---|---|---|---|---|---|

${\tilde{\Lambda}}_{CCC,1}$ | $0$ | $0.3812$ | $0.3812$ | $1$ | $0.6188$ | $2.3812$ | $0.2966$ |

${\tilde{\Lambda}}_{CCC,2}$ | $0$ | $0.8840$ | $0.8840$ | $1$ | $0.1160$ | $2.8840$ | $0.0748$ |

${\tilde{\Lambda}}_{CCC,3}$ | $0.4804$ | 1 | $0.5196$ | $0.5196$ | $0$ | $2.5196$ | $0.4050$ |

${\tilde{\Lambda}}_{CCC,4}$ | $0$ | $0.3370$ | $0.3370$ | $1$ | $0.6630$ | $2.3370$ | $0.0228$ |

${\tilde{\Lambda}}_{CCC,5}$ | $0$ | $0.0663$ | $0.0663$ | $1$ | $0.9337$ | $2.0663$ | $0.0414$ |

${\tilde{\Lambda}}_{CCC,WJR}$ | 0 | 0.3370 | 0.3370 | 1 | 0.9337 | ||

${\tilde{\Lambda}}_{CCC,SJR}$ | 0 | 0.0663 | 0.0663 | 1 | 0.6630 |

Criterion | ${\mathit{A}}^{--}$ | ${\mathit{A}}^{-}$ | $$ |
---|